Isn't it odd how such a small thing about asking a girl to go to a dance with you becomes such a big thing? Seriously.. EVERY girl is dying to go to EVERY dance. Yet a lot of them aren't able to, or have to go with a bunch of their chick-friends, because no guy will ask them. This is one situation where the "market" is in your favor - there are more girls who want to go to dances than there are guys willing to ask them. You're doing a HUGE favor to any girl you do choose to ask. Girls attach huge social status to having/not having a date to a dance - no girl will reject you unless she's already going with someone (oh well, not your fault) or she thinks she can do better (which if you apply the ideas in the DJ bible, won't happen). It's all good.
The reason a really simple thing like this becomes hard is because you think about it too much. Maybe you've heard of putting girls on a pedestal; you can put activities/events on a pedestal too. You attach all kinds of importance to a dance (like being cool, being able to get girls, etc.) so of course you'll be nervous about it - you've created a huge psychological stake for yourself. However, in reality, dances are NOT A BIG DEAL. You go to dinner with a person, probably in a group, then you go stand on a dance floor for a couple hours.
I went to all the mixers my sophomore year. Junior year, my school put doomfook-style regulations on how we could dance, so I only went to the christmas dance (didn't go to prom). I got so pissed at my date there that I almost walked out on her and ditched her there... I'm a senior now, so I'll have to go to prom this year, just to uphold appearances... But it still feels like more of a chore than a blast. This is the thing you're worrying over turning out badly?
About making yourself look dumb dancing... I used to have the same fear. There are a couple factors you have to take into account though.
1. 90% of the other people have no idea what they're doing either. I don't know anyone who has taken dance lessons (I'm sure they're out there, but they're not common). That means most everybody is on the same playing field as you are.
2. Dancing is really, really easy. Rap - move your feet around, move your hips around, somewhat related to the beat. You can also get really good at dancing to rap if you feel like practicing at home... There are threads on this. Rock - jump around. Slow - grab a girl. Her arms go over your shoulders. Your arms go around her waist. How close you dance to her depends on preference.
3. Nobody is watching. People aren't evaluating your skills. This is always what bugged me; "what if I look stupid??" This is an illogical fear... Everyone around you is doing the exact same thing, and they're too involved to be judging you. If anyone IS watching, it's probably because you're GOOD, not the other way around. Also, keep in mind that even if people ARE watching, and you totally suck, you're still doing better than all the guys who just sit on the edges of the floor doing nothing.. It's pathetic. Don't be like that.
This ended up a bit longer than I intended.. But I hope it was helpful. Being really shy sucks, good luck with turning that around.